The Laplacian is self-adjoint with respect to the scalar product on \(L^2(G)\)  that uses the Haar measure. One has to take this into account in order to obtain an operator of the Schrödinger form on \(\Gamma ^A\) . This is achieved by conjugation with a scalar factor \(\omega \)  [1]}
\(\omega (u_1,u_2) = 4(-1)^{2-d}\left(\sinh \frac{u_1}{2} \sinh \frac{u_2}{2}\right)^{2d-2}\coth \frac{u_1}{2}\coth \frac{u_2}{2} \left|\sinh ^{-2}\frac{u_1}{2}-\sinh ^{-2}\frac{u_2}{2}\right|^{d-2} .\) 
